How do I fix this weird lighting (normals?) problem?
As others have mentioned, Unity is probably getting confused about what direction the normals should be in. You need to change the topology of the large area that causes trouble. I think this would be the most logical choice:



Have you tried this combination?
To get to it, simply delete all the "wrong" faces, insert a face (F) in their place, and cut it up using the knife tool (K).



If this doesn't work, or if you're too lazy :), try adding a Bevel modifier with a small Width.